Key words and phrases: Jessen’s Inequality, Isotonic Linear Functionals

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http://jipam.vu.edu.au/

Volume 2, Issue 3, Article 36, 2001

ON A REVERSE OF JESSEN’S INEQUALITY FOR ISOTONIC LINEAR FUNCTIONALS

S.S. DRAGOMIR

SCHOOL OFCOMMUNICATIONS ANDINFORMATICS

VICTORIAUNIVERSITY OFTECHNOLOGY

PO BOX14428 MELBOURNECITYMC VICTORIA8001, AUSTRALIA. sever@matilda.vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html

Received 23 April, 2001; accepted 28 May, 2001.

Communicated by A. Lupas

ABSTRACT. A reverse of Jessen’s inequality and its version for m−Ψ−convex and M − Ψ−convex functions are obtained. Some applications for particular cases are also pointed out.

Key words and phrases: Jessen’s Inequality, Isotonic Linear Functionals.

2000 Mathematics Subject Classification. 26D15, 26D99.

1. INTRODUCTION

LetLbe a linear class of real-valued functionsg :E →Rhaving the properties (L1) f, g ∈Limply(αf +βg)∈Lfor allα, β ∈R;

(L2) 1∈L,i.e., iff₀(t) = 1,t∈E thenf₀ ∈L.

An isotonic linear functionalA:L→Ris a functional satisfying (A1) A(αf +βg) =αA(f) +βA(g)for allf, g ∈Landα, β ∈R. (A2) Iff ∈Landf ≥0, thenA(f)≥0.

The mappingAis said to be normalised if (A3) A(1) = 1.

Isotonic, that is, order-preserving, linear functionals are natural objects in analysis which enjoy a number of convenient properties. Thus, they provide, for example, Jessen’s inequality, which is a functional form of Jensen’s inequality (see [2] and [10]).

We recall Jessen’s inequality (see also [8]).

ISSN (electronic): 1443-5756

047-01

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Theorem 1.1. Letφ:I ⊆R→R(I is an interval), be a convex function andf :E →I such thatφ◦f,f ∈L. IfA:L→Ris an isotonic linear and normalised functional, then

(1.1) φ(A(f))≤A(φ◦f).

A counterpart of this result was proved by Beesack and Peˇcari´c in [2] for compact intervals I = [α, β].

Theorem 1.2. Letφ : [α, β]⊂R→Rbe a convex function andf :E →[α, β]such thatφ◦f, f ∈L. IfA:L→Ris an isotonic linear and normalised functional, then

(1.2) A(φ◦f)≤ β−A(f)

β−α φ(α) + A(f)−α β−α φ(β). Remark 1.3. Note that (1.2) is a generalisation of the inequality

(1.3) A(φ)≤ b−A(e1)

b−a φ(a) + A(e1)−a b−a φ(b)

due to Lupa¸s [9] (see for example [2, Theorem A]), which assumedE = [a, b],Lsatisfies (L1), (L2),A : L→ Rsatisfies (A1), (A2),A(1) = 1,φ is convex onE andφ ∈ L, e1 ∈ L, where e₁(x) =x,x∈[a, b].

The following inequality is well known in the literature as the Hermite-Hadamard inequality

(1.4) ϕ

a+b 2

≤ 1 b−a

Z b a

ϕ(t)dt ≤ ϕ(a) +ϕ(b)

2 ,

provided thatϕ : [a, b]→Ris a convex function.

Using Theorem 1.1 and Theorem 1.2, we may state the following generalisation of the Hermite-Hadamard inequality for isotonic linear functionals ([11] and [12]).

Theorem 1.4. Letφ: [a, b]⊂R→Rbe a convex function ande :E →[a, b]withe,φ◦e∈L.

IfA →Ris an isotonic linear and normalised functional, withA(e) = ^a+b₂ , then

(1.5) ϕ

a+b 2

≤A(φ◦e)≤ ϕ(a) +ϕ(b)

2 .

For other results concerning convex functions and isotonic linear functionals, see [3] – [6], [12] – [14] and the recent monograph [7].

2. THECONCEPTS OFm−Ψ−CONVEX ANDM−Ψ−CONVEX FUNCTIONS

Assume that the mappingΨ : I ⊆R→R(I is an interval) is convex onI andm ∈ R. We shall say that the mappingφ :I →Rism−Ψ−lower convex ifφ−mΨis a convex mapping onI (see [4]). We may introduce the class of functions

(2.1) L(I, m,Ψ) :={φ :I →R|φ−mΨ is convex onI}.

Similarly, for M ∈ R and Ψas above, we can introduce the class of M −Ψ−upper convex functions by (see [4])

(2.2) U(I, M,Ψ) :={φ:I →R|MΨ−φ is convex onI}.

The intersection of these two classes will be called the class of(m, M)−Ψ−convex functions and will be denoted by

(2.3) B(I, m, M,Ψ) :=L(I, m,Ψ)∩ U(I, M,Ψ).

Remark 2.1. IfΨ∈ B(I, m, M,Ψ), thenφ−mΨandMΨ−φare convex and then(φ−mΨ)+

(MΨ−φ)is also convex which shows that(M−m) Ψis convex, implying thatM ≥ m(as Ψis assumed not to be the trivial convex functionΨ (t) = 0,t∈I).

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The above concepts may be introduced in the general case of a convex subset in a real linear space, but we do not consider this extension here.

In [6], S.S. Dragomir and N.M. Ionescu introduced the concept of g−convex dominated mappings, for a mappingf :I →R. We recall this, by saying, for a given convex functiong : I →R, the functionf :I →Risg−convex dominated iffg+fandg−fare convex mappings onI. In [6], the authors pointed out a number of inequalities for convex dominated functions related to Jensen’s, Fuchs’, Peˇcarić’s, Barlow-Marshall-Proschan and Vasić-Mijalković results, etc.

We observe that the concept ofg−convex dominated functions can be obtained as a particular case from(m, M)−Ψ−convex functions by choosingm=−1,M = 1andΨ =g.

The following lemma holds (see also [4]).

Lemma 2.2. LetΨ, φ:I ⊆R→Rbe differentiable functions on ˚I andΨis a convex function on ˚I.

(i) Form∈R, the functionφ∈ L ˚I, m,Ψ iff

(2.4) m[Ψ (x)−Ψ (y)−Ψ⁰(y) (x−y)]≤φ(x)−φ(y)−φ⁰(y) (x−y) for allx, y ∈˚I.

(ii) ForM ∈R, the functionφ ∈ U ˚I, M,Ψ iff

(2.5) φ(x)−φ(y)−φ⁰(y) (x−y)≤M[Ψ (x)−Ψ (y)−Ψ⁰(y) (x−y)]

for allx, y ∈˚I.

(iii) For M, m ∈ RwithM ≥ m, the functionφ ∈ B ˚I, m, M,Ψ

iff both (2.4) and (2.5) hold.

Proof. Follows by the fact that a differentiable mappingh : I → Ris convex on ˚I iffh(x)−

h(y)≥h⁰(y) (x−y)for allx, y ∈˚I.

Another elementary fact for twice differentiable functions also holds (see also [4]).

Lemma 2.3. LetΨ, φ:I ⊆R→Rbe twice differentiable on ˚I andΨis convex on ˚I.

(i) Form∈R, the functionφ∈ L ˚I, m,Ψ iff

(2.6) mΨ⁰⁰(t)≤φ⁰⁰(t) for allt∈ ˚I.

(ii) ForM ∈R, the functionφ ∈ U ˚I, M,Ψ iff

(2.7) φ⁰⁰(t)≤MΨ⁰⁰(t) for allt∈ ˚I.

(iii) For M, m ∈ RwithM ≥ m, the functionφ ∈ B ˚I, m, M,Ψ

iff both (2.6) and (2.7) hold.

Proof. Follows by the fact that a twice differentiable function h : I → R is convex on ˚I iff

h⁰⁰(t)≥0for allt∈˚I.

We consider thep−logarithmic mean of two positive numbers given by

L_p(a, b) :=











a if b=a,

b^p+1−a^p+1 (p+ 1) (b−a)

¹_p

if a6=b andp∈R{−1,0}.

The following proposition holds (see also [4]).

Proposition 2.4. Letφ: (0,∞)→Rbe a differentiable mapping.

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(i) Form∈R, the functionφ∈ L((0,∞), m,(·)^p)withp∈(−∞,0)∪(1,∞)iff

(2.8) mp(x−y)

L^p−1_p−1(x, y)−y^p−1

≤φ(x)−φ(y)−φ⁰(y) (x−y) for allx, y ∈(0,∞).

(ii) ForM ∈R, the functionφ ∈ U((0,∞), M,(·)^p)withp∈(−∞,0)∪(1,∞)iff (2.9) φ(x)−φ(y)−φ⁰(y) (x−y)≤M p(x−y)

L^p−1_p−1(x, y)−y^p−1 for allx, y ∈(0,∞).

(iii) ForM, m∈RwithM ≥m, the functionφ∈ B((0,∞), M,(·)^p)withp∈(−∞,0)∪ (1,∞)iff both (2.8) and (2.9) hold.

The proof follows by Lemma 2.2 applied for the convex mappingΨ (t) = t^p,p∈(−∞,0)∪ (4,∞)and via some elementary computation. We omit the details.

The following corollary is useful in practice.

Corollary 2.5. Letφ: (0,∞)→Rbe a differentiable function.

(i) Form∈R, the functionφism−quadratic-lower convex (i.e., forp= 2) iff (2.10) m(x−y)² ≤φ(x)−φ(y)−φ⁰(y) (x−y)

for allx, y ∈(0,∞).

(ii) ForM ∈R, the functionφisM−quadratic-upper convex iff (2.11) φ(x)−φ(y)−φ⁰(y) (x−y)≤M(x−y)²

for allx, y ∈(0,∞).

(iii) Form, M ∈RwithM ≥m, the functionφis(m, M)−quadratic convex if both (2.10) and (2.11) hold.

The following proposition holds (see also [4]).

Proposition 2.6. Letφ: (0,∞)→Rbe a twice differentiable function.

(i) Form∈R, the functionφ∈ L((0,∞), m,(·)^p)withp∈(−∞,0)∪(1,∞)iff (2.12) p(p−1)mt^p−2 ≤φ⁰⁰(t) for allt∈(0,∞).

(ii) ForM ∈R, the functionφ ∈ U((0,∞), M,(·)^p)withp∈(−∞,0)∪(1,∞)iff (2.13) φ⁰⁰(t)≤p(p−1)M t^p−2 for allt∈(0,∞).

(iii) For m, M ∈ R with M ≥ m, the function φ ∈ B((0,∞), m, M,(·)^p) with p ∈ (−∞,0)∪(1,∞)iff both (2.12) and (2.13) hold.

As can be easily seen, the above proposition offers the practical criterion of deciding when a twice differentiable mapping is (·)^p−lower or (·)^p−upper convex and which weights the constantmandM are.

Corollary 2.7. Assume that the mappingφ: (a, b)⊆R→Ris twice differentiable.

(i) If inf

t∈(a,b)φ⁰⁰(t) =k >−∞, thenφis ^k₂−quadratic lower convex on(a, b) ; (ii) If sup

t∈(a,b)

φ⁰⁰(t) =K <∞, thenφis ^K₂−quadratic upper convex on(a, b).

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3. A REVERSE INEQUALITY

We start with the following result which gives another counterpart forA(φ◦f), as did the Lupa¸s-Beesack-Peˇcari´c result (1.2).

Theorem 3.1. Letφ : (α, β)⊆R→Rbe a differentiable convex function on(α, β),f :E → (α, β)such thatφ◦f,f,φ⁰◦f,φ⁰◦f·f ∈L. IfA:L→Ris an isotonic linear and normalised functional, then

0 ≤ A(φ◦f)−φ(A(f)) (3.1)

≤ A(φ⁰◦f ·f)−A(f)·A(φ⁰◦f)

≤ 1

4[φ⁰(β)−φ⁰(α)] (β−α) (ifα,βare finite).

Proof. Asφis differentiable convex on(α, β), we may write that

(3.2) φ(x)−φ(y)≥φ⁰(y) (x−y), for allx, y ∈(α, β), from where we obtain

(3.3) φ(A(f))−(φ◦f) (t)≥(φ⁰◦f) (t) (A(f)−f(t)) for allt∈E, as, obviously,A(f)∈(α, β).

If we apply to (3.3) the functionalA, we may write

φ(A(f))−A(φ◦f)≥A(f)·A(φ⁰◦f)−A(φ⁰◦f ·f), which is clearly equivalent to the first inequality in (3.1).

It is well known that the following Grüss inequality for isotonic linear and normalised functionals holds (see [1])

(3.4) |A(hk)−A(h)A(k)| ≤ 1

4(M −m) (N −n),

provided thath, k ∈ L,hk ∈ Land−∞ < m ≤ h(t) ≤M < ∞,−∞ < n ≤k(t) ≤ N <

∞, for allt∈E.

Taking into account that for finiteα,β we have α < f(t) < β withφ⁰ being monotonic on (α, β), we haveφ⁰(α)≤φ⁰◦f ≤φ⁰(β), and then by the Grüss inequality, we may state that

A(φ⁰◦f ·f)−A(f)·A(φ⁰◦f)≤ 1

4[φ⁰(β)−φ⁰(α)] (β−α)

and the theorem is completely proved.

The following corollary holds.

Corollary 3.2. Let φ : [a, b] ⊂˚I⊆ R→R be a differentiable convex function on ˚I. If φ, e₁, φ⁰, φ⁰ ·e₁ ∈ L(e₁(x) = x, x ∈ [a, b]) and A : L → R is an isotonic linear and normalised functional, then:

0 ≤ A(φ)−φ(A(e1)) (3.5)

≤ A(φ⁰·e₁)−A(e₁)·A(φ⁰)

≤ 1

4[φ⁰(b)−φ⁰(a)] (b−a). There are some particular cases which can naturally be considered.

(1) Let φ(x) = lnx, x > 0. If lnf, f, _f¹ ∈ Land A : L → Ris an isotonic linear and normalised functional, then:

(3.6) 0≤ln [A(f)]−A[ln (f)]≤A(f)A 1

f

−1,

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provided thatf(t)>0for allt∈E andA(f)>0.

If0< m≤f(t)≤M <∞,t∈E, then, by the second part of (3.1) we have:

(3.7) A(f)A

1 f

−1≤ (M −m)²

4mM (which is a known result).

Note that the inequality (3.6) is equivalent to

(3.8) 1≤ A(f)

exp [A[ln (f)]] ≤exp

A(f)A 1

f

−1

.

(2) Letφ(x) = exp (x),x∈R. Ifexp (f),f,f·exp (f)∈LandA:L→Ris an isotonic linear and normalised functional, then

0 ≤ A[exp (f)]−exp [A(f)]

(3.9)

≤ A[fexp (f)]−A(f) exp [A(f)]

≤ 1

4[exp (M)−exp (m)] (M −m) (if m≤f ≤M onE).

4. A FURTHER RESULT FORm−Ψ−CONVEX ANDM −Ψ−CONVEX FUNCTIONS

In [4], S.S. Dragomir proved the following inequality of Jessen’s type for m−Ψ−convex andM −Ψ−convex functions.

Theorem 4.1. LetΨ :I ⊆R→Rbe a convex function andf :E →I such thatΨ◦f,f ∈L andA :L→Rbe an isotonic linear and normalised functional.

(i) Ifφ∈ L(I, m,Ψ)andφ◦f ∈L, then we have the inequality (4.1) m[A(Ψ◦f)−Ψ (A(f))]≤A(φ◦f)−φ(A(f)).

(ii) Ifφ∈ U(I, M,Ψ)andφ◦f ∈L, then we have the inequality (4.2) A(φ◦f)−φ(A(f))≤M[A(Ψ◦f)−Ψ (A(f))].

(iii) Ifφ∈ B(I, m, M,Ψ)andφ◦f ∈L, then both (4.1) and (4.2) hold.

Corollary 4.2. LetΨ : I ⊆ R→Rbe a twice differentiable convex function on ˚I,f : E → I such thatΨ◦f,f ∈LandA:L→Rbe an isotonic linear and normalised functional.

(i) Ifφ:I →Ris twice differentiable andφ⁰⁰(t)≥mΨ⁰⁰(t),t∈˚I (wheremis a given real number), then (4.1) holds, provided thatφ◦f ∈L.

(ii) If φ : I → Ris twice differentiable andφ⁰⁰(t) ≤ MΨ⁰⁰(t), t ∈˚I (whereM is a given real number), then (4.2) holds, provided thatφ◦f ∈L.

(iii) Ifφ :I →Ris twice differentiable and mΨ⁰⁰(t)≤ φ⁰⁰(t) ≤ MΨ⁰⁰(t), t ∈˚I, then both (4.1) and (4.2) hold, providedφ◦f ∈L.

In [5], S.S. Dragomir obtained the following result of Lupa¸s-Beesack-Peˇcari´c type for m− Ψ−convex andM −Ψ−convex functions.

Theorem 4.3. LetΨ : [α, β]⊂R→Rbe a convex function andf :I →[α, β]such thatΨ◦f, f ∈LandA:L→Ris an isotonic linear and normalised functional.

(i) Ifφ∈ L(I, m,Ψ)andφ◦f ∈L, then we have the inequality (4.3) m

β−A(f)

β−α Ψ (α) + A(f)−α

β−α Ψ (β)−A(Ψ◦f)

≤ β−A(f)

β−α φ(α) + A(f)−α

β−α φ(β)−A(φ◦f).

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(ii) Ifφ∈ U(I, M,Ψ)andφ◦f ∈L, then (4.4) β−A(f)

β−α φ(α) + A(f)−α

β−α φ(β)−A(φ◦f)

≤M

β−A(f)

β−α Ψ (α) + A(f)−α

β−α Ψ (β)−A(Ψ◦f)

. (iii) Ifφ∈ B(I, m, M,Ψ)andφ◦f ∈L, then both (4.3) and (4.4) hold.

Corollary 4.4. LetΨ : I ⊆ R→Rbe a twice differentiable convex function on ˚I,f : E → I such thatΨ◦f,f ∈LandA:L→Ris an isotonic linear and normalised functional.

(i) Ifφ :I → Ris twice differentiable,φ◦f ∈Landφ⁰⁰(t)≥mΨ⁰⁰(t),t ∈˚I (wheremis a given real number), then (4.3) holds.

(ii) Ifφ:I →Ris twice differentiable,φ◦f ∈Landφ⁰⁰(t)≤MΨ⁰⁰(t),t∈˚I (wheremis a given real number), then (4.4) holds.

(iii) IfmΨ⁰⁰(t)≤φ⁰⁰(t)≤MΨ⁰⁰(t),t ∈˚I, then both (4.3) and (4.4) hold.

We now prove the following new result.

Theorem 4.5. LetΨ :I ⊆ R→Rbe differentiable convex function andf : E → I such that Ψ◦f,Ψ⁰◦f,Ψ⁰◦f·f,f ∈LandA:L→Rbe an isotonic linear and normalised functional.

(i) Ifφis differentiable,φ ∈ L ˚I, m,Ψ

andφ◦f,φ⁰◦f,φ⁰◦f ·f ∈L, then we have the inequality

(4.5) m[A(Ψ⁰◦f ·f) + Ψ (A(f))−A(f)·A(Ψ⁰ ◦f)−A(Ψ◦f)]

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f). (ii) Ifφis differentiable,φ∈ U ˚I, M,Ψ

andφ◦f,φ⁰◦f,φ⁰◦f·f ∈L, then we have the inequality

(4.6) A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f)

≤M[A(Ψ⁰◦f ·f) + Ψ (A(f))−A(f)·A(Ψ⁰◦f)−A(Ψ◦f)]. (iii) Ifφis differentiable,φ∈ B ˚I, m, M,Ψ

andφ◦f,φ⁰◦f,φ⁰◦f·f ∈L, then both (4.5) and (4.6) hold.

Proof. The proof is as follows.

(i) As φ ∈ L(I, m,Ψ), then φ − mΨ is convex and we can apply the first part of the inequality (3.1) forφ−mΨgetting

(4.7) A[(φ−mΨ)◦f]−(φ−mΨ) (A(f))

≤A

(φ−mΨ)⁰◦f ·f

−A(f)A (φ−mΨ)⁰◦f . However,

A[(φ−mΨ)◦f] = A(φ◦f)−mA(Ψ◦f), (φ−mΨ) (A(f)) = φ(A(f))−mΨ (A(f)), A

(φ−mΨ)⁰◦f·f

= A(φ⁰◦f ·f)−mA(Ψ⁰◦f ·f) and

A (φ−mΨ)⁰◦f

=A(φ⁰◦f)−mA(Ψ⁰◦f) and then, by (4.7), we deduce the desired inequality (4.5).

(ii) Goes likewise and we omit the details.

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(iii) Follows by(i)and(ii).

The following corollary is useful in practice,

Corollary 4.6. LetΨ : I ⊆R → Rbe a twice differentiable convex function on ˚I, f : E → I such thatΨ◦f,Ψ⁰◦f,Ψ⁰◦f·f,f ∈LandA:L→Rbe an isotonic linear and normalised functional.

(i) Ifφ :I →Ris twice differentiable,φ◦f,φ⁰◦f,φ⁰◦f ·f ∈ Landφ⁰⁰(t)≥mΨ⁰⁰(t), t ∈˚I, (wheremis a given real number), then the inequality (4.5) holds.

(ii) With the same assumptions, but if φ⁰⁰(t) ≤ MΨ⁰⁰(t), t ∈˚I, (where M is a given real number), then the inequality (4.6) holds.

(iii) IfmΨ⁰⁰(t)≤φ⁰⁰(t)≤MΨ⁰⁰(t),t ∈˚I, then both (4.5) and (4.6) hold.

Some particular important cases of the above corollary are embodied in the following proposition.

Proposition 4.7. Assume that the mappingφ:I ⊆R→Ris twice differentiable on ˚I.

(i) Ifinf

t∈˚Iφ⁰⁰(t) = k >−∞, then we have the inequality (4.8) 1

2k

A f²

−[A(f)]²

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f), provided thatφ◦f,φ⁰◦f,φ⁰◦f ·f, f² ∈L.

(ii) Ifsup

t∈˚I

φ⁰⁰(t) = K <∞, then we have the inequality (4.9) A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f)

≤ 1 2K

A f²

−[A(f)]² . (iii) If−∞< k≤φ⁰⁰(t)≤K <∞,t∈˚I, then both (4.8) and (4.9) hold.

The proof follows by Corollary 4.6 applied forΨ (t) = ¹₂t² andm =k,M =K. Another result is the following one.

Proposition 4.8. Assume that the mappingφ:I ⊆(0,∞)→Ris twice differentiable on ˚I. Let p∈(−∞,0)∪(1,∞)and defineg_p :I →R,g_p(t) = φ⁰⁰(t)t^2−p.

(i) Ifinf

t∈˚Ig_p(t) =γ >−∞, then we have the inequality

(4.10) γ

p(p−1)

(p−1) [A(f^p)−[A(f)]^p]−pA(f)

A f^p−1

−[A(f)]^p−1

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f), provided thatφ◦f,φ⁰◦f,φ⁰◦f ·f, f^p, f^p−1 ∈L.

(ii) Ifsup

t∈˚I

g_p(t) = Γ<∞, then we have the inequality

≤ Γ p(p−1)

(p−1) [A(f^p)−[A(f)]^p]−pA(f)

A f^p−1

−[A(f)]^p−1 . (iii) If−∞< γ≤g_p(t)≤Γ<∞,t∈˚I, then both (4.10) and (4.11) hold.

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(i) We have for the auxiliary mappingh_p(t) =φ(t)− _p(p−1)^γ t^p that h⁰⁰_p(t) = φ⁰⁰(t)−γt^p−2 =t^p−2 t^2−pφ⁰⁰(t)−γ

= t^p−2(g_p(t)−γ)≥0.

That is, h_p is convex or, equivalently,φ ∈ L

I,_p(p−1)^γ ,(·)^p

. Applying Corollary 4.6, we get

γ p(p−1)

pA(f^p) + [A(f)]^p−pA(f)A f^p−1

−A(f^p)

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f), which is clearly equivalent to (4.10).

(ii) Goes similarly.

The following proposition also holds.

Proposition 4.9. Assume that the mapping φ : I ⊆ (0,∞) → Ris twice differentiable on ˚I.

Definel(t) = t²φ⁰⁰(t),t∈I.

(i) Ifinf

t∈˚Il(t) =s >−∞, then we have the inequality (4.12) s

A(f)A 1

f

−1−(ln [A(f)]−A[ln (f)])

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f), provided thatφ◦f, φ⁻¹◦f, φ⁻¹◦f·f, ¹_f,lnf ∈LandA(f)>0.

(ii) Ifsup

t∈˚I

l(t) =S < ∞, then we have the inequality

≤S

A(f)A 1

f

−1−(ln [A(f)]−A[ln (f)])

. (iii) If−∞< s≤l(t)≤S <∞fort ∈˚I, then both (4.12) and (4.13) hold.

(i) Define the auxiliary functionh(t) =φ(t) +slnt. Then h⁰⁰(t) =φ⁰⁰(t)− s

t² = 1

t² φ⁰⁰(t)t²−s

≥0

which shows that his convex, or, equivalently, φ ∈ L(I, s,−ln (·)). Applying Corol- lary 4.6, we may write

s

−A(1)−lnA(f) +A(f)A 1

f

+A(ln (f))

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f), which is clearly equivalent to (4.12).

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Finally, the following result also holds.

Proposition 4.10. Assume that the mappingφ : I ⊆ (0,∞) → Ris twice differentiable on ˚I.

DefineI˜(t) =tφ⁰⁰(t),t ∈I.

(i) Ifinf

t∈˚I

I˜(t) = δ >−∞, then we have the inequality (4.14) δA(f) [ln [A(f)]−A(ln (f))]

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰ ◦f)−A(φ◦f), provided thatφ◦f, φ0 ◦f, φ0 ◦f·f,lnf, f ∈LandA(f)>0.

(ii) Ifsup

t∈˚I

I˜(t) = ∆ <∞, then we have the inequality

≤∆A(f) [ln [A(f)]−A(ln (f))]. (iii) If−∞< δ ≤I˜(t)≤∆<∞fort∈˚I, then both (4.14) and (4.15) hold.

(i) Define the auxiliary mappingh(t) =φ(t)−δtlnt,t∈I. Then h⁰⁰(t) =φ⁰⁰(t)− δ

t = 1

t² [φ⁰⁰(t)t−δ] = 1 t

hI˜(t)−δi

≥0

which shows thathis convex or equivalently,φ∈ L(I, δ,(·) ln (·)). Applying Corollary 4.6, we get

δ[A[(lnf+ 1)f] +A(f) lnA(f)−A(f)A(lnf+ 1)−A(flnf)]

≤A(φ⁰◦f·f) +φ(A(f))−A(f)·A(φ⁰◦f)−A(φ◦f) which is equivalent with (4.14).

5. SOME APPLICATIONSFOR BULLEN’S INEQUALITY

The following inequality is well known in the literature as Bullen’s inequality (see for example [7, p. 10])

(5.1) 1

b−a Z b

a

φ(t)dt ≤ 1 2

φ(a) +φ(b)

2 +φ

a+b 2

,

provided thatφ : [a, b]→Ris a convex function on[a, b]. In other words, as (5.1) is equivalent to:

(5.2) 0≤ 1

b−a Z b

a

φ(t)dt−φ

a+b 2

≤ φ(a) +φ(b)

2 − 1

b−a Z b

a

φ(t)dt we can conclude that in the Hermite-Hadamard inequality

(5.3) φ(a) +φ(b)

2 ≥ 1

b−a Z b

a

φ(t)dt≥φ

a+b 2

the integral mean _b−a¹ Rb

aφ(t)dtis closer toφ ^a+b₂

than to ^φ(a)+φ(b)₂ .

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Using some of the results pointed out in the previous sections, we may upper and lower bound the Bullen difference:

B(φ;a, b) := 1 2

φ(a) +φ(b)

2 +φ

a+b 2

− 1 b−a

Z b a

φ(t)dt

(which is positive for convex functions) for different classes of twice differentiable functionsφ.

Now, if we assume that A(f) := _b−a¹ Rb

a f(t)dt, then for f = e,e(x) = x, x ∈ [a, b], we have, for a differentiable functionφ, that

A(φ⁰◦f ·f) +φ(A(f))−A(f)·A(φ⁰◦f)−A(φ◦f)

= 1

b−a Z b

a

xφ⁰(x)dx+φ

a+b 2

− a+b

2 · 1

b−a Z b

a

φ⁰(x)dx− 1 b−a

Z b a

φ(x)dx

= 1

b−a

bφ(b)−aφ(a)− Z b

a

φ(x)dx

+φ

a+b 2

− a+b

2 ·φ(b)−φ(a) b−a − 1

b−a Z b

a

φ(x)dx

= φ(a) +φ(b)

2 +φ

a+b 2

− 2 b−a

Z b a

φ(x)dx

= 2B(φ;a, b).

a) Assume thatφ : [a, b] ⊂ R→Ris a twice differentiable function satisfying the property that−∞< k ≤φ⁰⁰(t)≤K <∞. Then by Proposition 4.7, we may state the inequality

(5.4) 1

48(b−a)²k ≤B(φ;a, b)≤ 1

48(b−a)²K.

This follows by Proposition 4.7 on taking into account that 1

b−a Z b

a

x²dx− 1

b−a Z b

a

xdx ²

= (b−a)² 12 .

b) Now, assume that the twice differentiable function φ : [a, b] ⊂ (0,∞) → R satisfies the property that −∞ < γ ≤ t^2−pφ⁰⁰(t) ≤ Γ < ∞, t ∈ (a, b), p ∈ (−∞,0)∪(1,∞). Then by Proposition 4.8 and taking into account that

A(f^p)−(A(f))^p = 1 b−a

Z b a

x^pdx− 1

b−a Z b

a

xdx ^p

= L^p_p(a, b)−A^p(a, b), and

A f^p−1

−(A(f))^p−1 =L^p−1_p−1(a, b)−A^p−1(a, b),

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we may state the inequality γ

p(p−1)

(p−1)

L^p_p(a, b)−A^p(a, b)

−pA(a, b)

L^p−1_p−1(a, b)−A^p−1(a, b) (5.5)

≤B(φ;a, b)

≤ Γ p(p−1)

(p−1)

L^p_p(a, b)−A^p(a, b)

−pA(a, b)

L^p−1_p−1(a, b)−A^p−1(a, b) .

c) Assume that the twice differentiable functionφ: [a, b]⊂(0,∞)→Rsatisfies the property that−∞< s≤t²φ⁰⁰(t)≤S <∞,t ∈(a, b), then by Proposition 4.9, and taking into account that

A(f)A f⁻¹

−1−ln [A(f)] +Aln (f) = A(a, b)

L(a, b) −1−lnA(a, b) +I(a, b)

= ln

I(a, b) A(a, b)·exp

A(a, b)−L(a, b) L(a, b)

, we get the inequality

s 2ln

I(a, b) A(a, b)·exp

A(a, b)−L(a, b) L(a, b)

(5.6)

≤B(φ;a, b)

≤ S 2 ln

I(a, b) A(a, b) ·exp

A(a, b)−L(a, b) L(a, b)

.

d) Finally, ifφsatisfies the condition−∞ < δ≤tφ⁰⁰(t)≤∆<∞, then by Proposition 4.10, we may state the inequality

(5.7) δA(a, b) ln

A(a, b) I(a, b)

≤B(φ;a, b)≤∆A(a, b) ln

A(a, b) I(a, b)

.

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